Please solve these problems according to the method given in their respective heading.

problem - 4, 7

1-17 GENERAL SOLUTION 14. (x2D2 + xD - 41)y = 1/x2 Solve the given nonhomogeneous ODE by variation of 15. (D2 + D)y = sec x - 10 sin 5x parameters or undetermined coefficients. Give a general 16. (x2D2 + xD + (x2 - 1)1)y = x3/2 cos x. solution. (Show the details of your work.) Hint. To find y1, )2 set y = ux-1/2 1. y" + y = csc x 17. (x2D2 + XD + (x2 - bly = x3/2 sin x. 2. y" - 4y' + 4y = x2e* Hint: As in Prob. 16. 3. x2y" - 2xy' + 2y = x3 cos x 18. TEAM PROJECT. Comparison of Methods. The 4. y" - 2y' + y = ex sin x undetermined-coefficient method should be used 5. y" + y = tan .x whenever possible because it is simpler. Compare it 6. x2y" - xy' + y = x In ixl with the present method as follows. 7. y" + y = cos x + sec x (a) Solve y" + 2y' - 15y = 17 sin 5x by both 8. y" - 4y' + 4y = 12e2*/x4 methods, showing all details, and compare. 9. (D2 - 2D + D)y = x2+ x-20* (b) Solve y" + 9y = 1 + 12, 1 = sec 3x, 10. (D2 - D)y = 1/cosh x 12 = sin 3x by applying each method to a suitable function on the right. 11. (D2 + 41)y = cosh 2x (c) Invent an undetermined-coefficient method for 12. (x2D2 + xD - 40)y = 3x-1 + 3x nonhomogeneous Euler-Cauchy equations by 13. (x2D2 - 2xD + 21)y = x3 sin x experimenting.1-14 GENERAL SOLUTIONS OF 16. y" - 3y' + 2.25y = 27(x2 - x), NONHOMOGENEOUS ODES y(0) = 20, y (0) = 30 Find a (real) general solution. Which rule are you using? 17. y" + 0.2y' + 0.26y = 1.220.5x (Show each step of your calculation.) y(0) = 3.5. y (0) = 0.35 1. y" + 3y' + 2y = 30e2z 18. y" - 2y' = 12e2x - 8e-27 2. y" + 4y' + 3.75y = 109 cos 5x y(0) = -2, y'(0) = 12 3. y" " - 16y = 19.2ez + 60ex 19. y" - y' - 12y = 144x3 + 12.5, 4. y" " + 9y = cos x + cos 3x y(0) = 5, y (0) = -0.5 "" + y' y' - 6y = 6x3 - 3x2 + 12x 20. y" + 2y' + 10y = 17 sin x - 37 sin 3x, 5. y "+ 4y' ' + 4y = e-2% sin 2x y(0) = 6.6, y'(0) = -2.2 6. y 7. y #" + 6y' + 73y = 80e" cos 4x 21. WRITING PROJECT. Initial Value Problem. Write 8. y" v" + 10y' + 25y = 100 sinh 5.x out all the details of Example 3 in your own words. 9. y" - 0.16y = 32 cosh 0.4x Discuss Fig. 51 in more detail. Why is it that some of 10. y" + 4y' + 6.25y = 3.125(x + 1)2 the "half-waves" do not reach the dashed curves, 11. y" + 1.44y = 24 cos 1.2x whereas others preceding them (and, of course, all later ones) excede the dashed curves? 12. y" + 9y = 18x + 36 sin 3x 22. TEAM PROJECT. Extensions of the Method of 13. y" + 4y' + 5y = 25x2 + 13 sin 2x Undetermined Coefficients. (a) Extend the method 14. y" + 2y' + y = 2x sin x to products of the function in Table 2.1. (b) Extend 15-20 INITIAL VALUE PROBLEMS FOR the method to Euler-Cauchy equations. Comment on NONHOMOGENEOUS ODEs the practical significance of such extensions. 23. CAS PROJECT. Structure of Solutions of Initial Solve the initial value problem. State which rules you are using. Show each step of your calculation in detail. Value Problems. Using the present method, find, graph, and discuss the solutions y of initial value problems of 15. y" + 4y = 16 cos 2x, y(0) = 0. y'(0) = 0 your own choice. Explore effects on solutions caused by